Skip to main content

Individualised Life Tables

Investigating Dynamics of Health, Work and Cohabitation in the UK

Abstract

A life table is a table which shows, at each age, the probability that a person in a given population will die before their next birthday. It can be used to calculate life expectancy and healthy life expectancy for people of different ages. In this work, using longitudinal datasets and panel data methods, we produce life tables for different subgroups of the population, defined according to cohabitation status, employment and other factors. As a first step, we estimate the dynamics of factors which are of particular importance in people’s lives: health, labour market participation, cohabitation and mortality. The significance of these variables is twofold: they determine the well-being of individuals, but the variables also determine the resources available to the individuals in times of ill health. Using the British Household Panel Survey, we analyse the extent to which these variables are influenced by one another, and by exogenous factors such as education and ethnicity. Estimating a system of probit models using simulation techniques, we are able to distinguish the effects of the exogenous and endogenous variables from state dependence and unobserved heterogeneity. We also correct for attrition and the initial conditions problem. We estimate time trends in mortality, health and other dependent variables to investigate whether a compression of morbidity has occurred in the recent past. Finally, the parameter estimates are used to simulate life tables for various sub-groups in the population and compare measures of life expectancy and healthy life expectancy for different groups.

This is a preview of subscription content, access via your institution.

Notes

  1. 1.

    \(\left\lceil x\right\rceil \) denotes the value of x rounded upwards to the next integer. Hence, we order the columns and rows in the covariance matrix first according to year and then according to equation.

  2. 2.

    We provide an example of such a life table in Appendix “Example of a Life Table”.

  3. 3.

    We do not report here how the first period error terms are derived. In order to obtain these, we use a simulation algorithm similar to that of the maximum likelihood procedure detailed in the main text. This way we take into account that the conditional distribution of v depends on the starting value of the dependent variables of the model.

  4. 4.

    \(\left\lceil x\right\rceil \) denotes the value of x rounded upwards to the next integer.

References

  1. Alder, J., Mayhew, L., Moody, S., Morris, R., & Shah, R. (2005). The chronic disease burden – An analysis of health risks and health care usage. Cass Business School, Faculty of Actuarial Science, 56 pp.

  2. Bajekal, M. (2002). Care homes and their residents. London: The Stationery Office.

    Google Scholar 

  3. Bebbington, A. C., & Darton, R. A. (1996). Healthy life expectancy in England and Wales: Recent evidence. PSSRU Discussion Paper 1205.

  4. Bebbington, A., & Comas-Herrera, A. (2000). Healthy life expectancy: Trends to 1998, and the implications for long term care costs. PSSRU Discussion Paper 1695 (London School of Economics).

  5. Blake, D. P., & Mayhew, L. D. (2006). On the sustainability of the UK State pension system in the light of population ageing and declining fertility. Economic Journal, 116(512), F286–F305.

    Article  Google Scholar 

  6. Bone, M., Bebbington, A. C., Jagger, C., Morgan, K., & Nicolaas, G. (1995). Health expectancy and its uses. London: HMSO.

    Google Scholar 

  7. Börsch-Supan, A., & Hajivassiliou, V. A. (1993). Smooth unbiased multivariate probability simulators for maximum likelihood estimation of limited dependent variables models. Journal of Econometrics, 58, 347–68.

    Article  Google Scholar 

  8. Bound, J., Schoenbaum, M., Stinebrickner, T. R., & Waidmann, T. (1999). The dynamic effects of health on the labor force transitions of older workers. Labour Economics, 6(2), 179–202.

    Article  Google Scholar 

  9. Brown, S. L. (2000). The effect of union type on psychological wellbeing: Depression cohabitants versus marrieds. Journal of Health and Social Behaviour, 41, 241–255.

    Article  Google Scholar 

  10. Butt, Z., Haberman, S., Verrall, R., & Wass, V. (2008). Calculating compensation for loss of future earnings: estimating and using work life expectancy. Journal of the Royal Statistical Society. Series A, 171(4), 1–37.

    Google Scholar 

  11. Cheung, Y. B. (2000). Marital status and mortality in British women: A longitudinal study. International Journal of Epidemiology, 29, 93–99.

    Article  Google Scholar 

  12. Disney, R., Emmerson, C., & Wakefield, M. (2006). Ill health and retirement in Britain: A panel data-based analysis. Journal of Health Economics, 25, 621–649.

    Article  Google Scholar 

  13. Domeij, D., & Johannesson, M. (2006). Consumption and health. Contributions to Macroeconomics, 6(1), 1314–1314.

    Google Scholar 

  14. French, E. (2005). The effects of health, wealth, and wages on labour supply and retirement behaviour. Review of Economic Studies, 72, 397–427.

    Article  Google Scholar 

  15. Fries, J. (1980). Aging, natural death and the compression of morbidity. New England Journal of Medicine, 303, 130–5.

    Article  Google Scholar 

  16. Fuchs, V. R. (2004). Reflections on the socio-economic correlates of health. Journal of Health Economics, 23(4), 629–36.

    Article  Google Scholar 

  17. Geweke, J. (1989). Efficient simulation from the multivariate normal distribution subject to linear inequality constraints and the evaluation of constraint probabilities. Duke University, Durham, N.C: Mimeo.

  18. Goldman, N., Korenman, S., & Weinstein, R. (1995). Marital status and health among the elderly. Social Science and Medicine, 40, 1717–1730.

    Article  Google Scholar 

  19. Government Actuary’s Department (2006). Period and cohort expectation of life tables. 2006-based projections; dataset available at www.gad.gov.uk.

  20. Gruenberg, E. M. (1977). The failures of success. Milbank Memorial Foundation Quarterly/Health and Society, 55, 3–24.

    Article  Google Scholar 

  21. Heckman, J. J. (1981). The incidentical parameters problem and the problem of initial conditions in estimating a discrete-time data stochastic process. In C. F. Manski, & D. McFadden (Eds.), Structural analysis of discrete data with econometric applications (Chapter 4, pp. 179–95). Cambridge: MIT.

    Google Scholar 

  22. Heyma, A. (2004). A structural dynamic analysis of retirement behavior in the Netherlands. Journal of Applied Econometrics, 19(6), 739–759.

    Article  Google Scholar 

  23. Karlsson, M., Mayhew, L., & Rickayzen, B. (2007). Long term care financing in 4 OECD countries: Fiscal burden and distributive effects. Health Policy, 80, 107–34.

    Article  Google Scholar 

  24. Karlsson M., Mayhew, L., Plumb, R., & Rickayzen, B. (2006). Future costs for long-term care. Cost projections for long-term care for older people in the United Kingdom. Health Policy, 75, 187–213.

    Article  Google Scholar 

  25. Lillard, L. A., & Panis, C. W. A. (1996). Marital status and mortality: The role of health. Demography, 33, 313–327.

    Article  Google Scholar 

  26. Manton, K. G. (1987). Response to an introduction to the compression of morbidity. Gerontologica Perspecta, 1, 23–30.

    Google Scholar 

  27. Mayhew, L. (2002). The neighbourhood health economy: A systematic approach to health and social risks at neighbourhood level. Actuarial Research paper No. 144, Cass Business School.

  28. United Nations, Population Division of the Department of Economic and Social Affairs (2006). World population prospects: The 2006 revision.

  29. Wilde, J. (2000). Identification of multiple equation probit models with endogenous dummy regressors. Economics Letters, 69(1), 309–12.

    Article  Google Scholar 

  30. Wilson, C. M., & Oswald, A. J. (2005). How does marriage affect physical and psychological health? A survey of the longitudinal evidence. IZA Discussion Paper No 1619.

  31. Wong, L. (2003). Structural estimation of marriage models. Journal of Labor Economics, 21(3), 699–727.

    Article  Google Scholar 

  32. Wooldridge, J. (2000). The initial conditions problem in dynamic nonlinear panel data models with unobserved heterogeneity. Mimeo, University of Michigan.

  33. Wooldridge, J. M. (2002). Econometric analysis of cross-section and panel data. Cambridge: MIT.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Martin Karlsson.

Additional information

This research was supported by a grant from the EPSRC.

Appendices

Appendix

Simulating a Population Based on Model Estimates

The maximum likelihood procedure has provided us with parameter estimates for the econometric model. For simplicity, we partition these parameters into four groups: one is denoted β, one is denoted d the third one denoted θ, and the fourth one is denoted κ. The parameter vector θ contains all parameters related to the initial conditions problem, i.e. the parameters θ 0, θ 1 and θ 2 from Eq. 7. Hence, this parameter vector relates to all variables which remain constant over the projection period. The parameter vector d contains all parameters related to state dependence. The parameter vector β contains all parameters related to time-varying exogenous variables such as age and time, and the parameter vector κ contains the parameters of the covariance matrix of the error terms (Σ; we have denoted these parameters ρ, σ and ω in the paper).

The different sets of parameters are outlined in Table 19 below.

Table 19 Parameters used in simulation

Obviously, the first three sets of parameter vectors contain parameters for each of the four estimating equations—hence, we can define vectors β a , d a and θ a for the parameters of the survival equation, and similarly for the other three equations.

Simulating a Subpopulation

We determine a sample size N, in this case 10,000. Assuming a maximum life length of T = 100 years from the start year (since we focus on 50-year olds, this is reasonable), we need to simulate a matrix of error terms and then make sure they have the appropriate correlations with each other (determined by Ω).

Simulating Error Terms

First, we simulate a matrix of standard normals:

$$ v=F^{-1}\left( u\right) $$

where u is a 4T×N matrix with each element \(u_{ij}\backsim U\left( 0,1\right) \) and \(F^{-1}\left( \cdot \right) \) is the inverse of the cdf of a standard normal distribution. Obviously, v is also 4T×N and the observations are iid with mean zero and variance 1.Footnote 3

Next, we build the 4T×4T covariance matrix Σ. This matrix is defined as

$$ \Sigma _{ij}=\sigma _{ab}+\rho _{a}^{\left\vert t-s\right\vert }\frac{\sqrt{ 1-\rho _{a}^{2}}\sqrt{1-\rho _{b}^{2}}}{1-\rho _{a}\rho _{b}}\omega _{ab} $$

whereFootnote 4 \(a=\left\lceil \frac{i}{4}\right\rceil \) and \( b=\left\lceil \frac{j}{4}\right\rceil \) identify the corresponding estimating equation,

$$ t=i-4\left( a-1\right) $$

and

$$ s=j-4\left( b-1\right) $$

identify the corresponding years. Hence, the element Σ ij tells us how the error term at time t in equation a is correlated with the error term at time s in equation b. The parameter σ ab is the covariance of the individual effect in equation a with the corresponding individual effect in equation b. Likewise, ω ab is the covariance of shocks in equation a with shocks in equation b. Whenever a = b, the corresponding variance is included.

Since Σ is positive definite and symmetric, we can carry out Cholesky decomposition. Hence, we define the 4T×4T matrix L as the lower diagonal Cholesky factor of the covariance matrix Σ:

$$ L\cdot L^{\prime }=\Sigma . $$

Then, premultiplying the matrix of simulated error terms v by the Cholesky factor,

$$ e=\left( Lv\right) ^{\prime } $$

we get a N×4T matrix of error terms, distributed according to \( N\left( 0,\Sigma \right) \). Next, we partition the matrix e so that we get one N×T matrix for each estimating equation. We denote this matrices e a, e w, e c and e h for convenience. They are derived from the oringinal matrix according to the equation

$$ e^{a}=e\left( \begin{array}{c} 1 \\ 0 \\ 0 \\ 0 \end{array} \right) \otimes I_{T} $$

and similarly for the four other equations.

Determining the Individual Effect

Next, we need to determine the individual effect. The deterministic part (see Eq. 7) is the same for all individuals in a subgroup. A subgroup is characterised by their values of the independent variables which remain constant over time as well as the initial state, represented by the variables W 0, C 0 and H 0. If we denote all other independent variables which remain constant (i.e. education, sex, ethnicity) Z, we can define the vector

$$ G=\left( \begin{array}{cccccccc} W_{0} &\phantom{0} C_{0} &\phantom{0} H_{0} &\phantom{0} W_{0}C_{0} &\phantom{0} W_{0}H_{0} &\phantom{0} C_{0}H_{0} &\phantom{0} W_{0}C_{0}H_{0} &\phantom{0} Z \end{array} \right) . $$

Then, the deterministic part of the individual effect in equation j (\(j\in \left\{ a,w,c,h\right\} \)) can be defined as

$$ \alpha _{j}=G\theta _{j} $$

where we have suppressed the random part of α from Eq. 7 since it is included in the matrix e. Also, since all individuals in a certain subgroup have the same individual effects α j (again, ignoring the random term), we suppressed the individual index i used in Eq. 7.

Simulating Outcomes

Having defined the individual effect α j , and constructed the matrix of error terms e, it is straightforward to simulate a population. This is done recursively, starting in year 1 and calculating the current state in all dimensions \(\left( A,W,C,H\right) \) for all simulated individuals, then moving on to the next year. Hence, we use the following procedure:

$$ A_{it}=A_{i,t-1}\cdot 1\left[ \alpha _{a}+X_{t}\beta _{a}+\left[ \begin{array}{ccc} W_{i,t-1} &\phantom{0} C_{i,t-1} &\phantom{0} H_{i,t-1} \end{array} \right] d_{a}+e_{it}^{a}\geq 0\right] $$

where A it takes on the value 1 if simulated individual i survives to period t; X t represents exogenous variables changing over time (age, time trend) and \(1\left[ \cdot \right] \) is the indicator function, taking on the value 1 whenever the expression in the square brackets is true.

For the other dependent variables, we follow the same procedure; also taking into account that individuals need to be alive to be working, cohabiting and healthy. Hence,

$$ W_{it}=A_{it}\cdot 1\left[ \alpha _{w}+X_{t}\beta _{w}+\left[ \begin{array}{ccc} W_{i,t-1} &\phantom{0} C_{i,t-1} &\phantom{0} H_{i,t-1} \end{array} \right] d_{w}+e_{it}^{w}\geq 0\right] $$

and then, for the remaining two, we also add simultaneously determined variables (such as W it ); hence:

$$ C_{it}=A_{it}\cdot 1\left[ \alpha _{c}+X_{t}\beta _{c}+\left[ \begin{array}{cccc} W_{i,t-1} &\phantom{0} C_{i,t-1} &\phantom{0} H_{i,t-1} &\phantom{0} W_{it} \end{array} \right] d_{c}+e_{it}^{c}\geq 0\right] $$
$$ H_{it}=A_{it}\cdot 1\left[ \alpha _{h}+X_{t}\beta _{h}+\left[ \begin{array}{ccccc} W_{i,t-1} &\phantom{0} C_{i,t-1} &\phantom{0} H_{i,t-1} &\phantom{0} W_{it} &\phantom{0} C_{it} \end{array} \right] d_{h}+e_{it}^{h}\geq 0\right] . $$

Hence, at the end of this exercise, we have four N×T matrices of simulated outcomes for the four variables A, W, C and H. This means that we can easily obtain life expectancy measures by simple matrix manipulations. For example:

$$ LE=\frac{\sum\limits_{i=1}^{N}\sum\limits_{t=1}^{T}A_{it}}{N}+0.5. $$

Similarly, healthy life expectancy can be calculated as:

$$ HLE=\frac{\sum\limits_{i=1}^{N}\sum\limits_{t=1}^{T}H_{it}}{N}+0.5H_{0}. $$

And the same goes for other combinations of the dependent variables, such as working healthy life expectancy etc.

Analysing the Effects of Changing Status

Next, we want to analyse the effect of moving an individual from a certain starting state to another one, without changing other characteristics. Obviously, the difference in, say, life expectancy between two groups i and j (which we call ‘gap’ in the paper) is simply the difference between the two:

$$ \Delta LE=LE_{i}-LE_{j}. $$

However, when we consider moving an individual from one state to another, we want to take into account the fact that they can be assumed to be different from individuals in the destination category—and this is arguably the reason why they were actually in a different category at the outset. This differences between individuals belonging to different groups are captured by the individual effect α i in our model. Hence, when we analyse the effect of moving an individual, we want to calculate counterfactual outcomes, based on a simulation where we keep α i constant but change the starting position in accordance with the destination category.

Consider the survival equation above. In this new setting, the ‘counterfactual’ survival in period 1 would be

$$ A_{i1}^{\prime }=1\left[ \alpha _{a}+X_{t}\beta _{a}+\left[ \begin{array}{ccc} W_{0}^{\prime } &\phantom{0} C_{0}^{\prime } &\phantom{0} H_{0}^{\prime } \end{array} \right] d_{a}+e_{i1}^{a}\geq 0\right] $$

where α a  =  a is determined according the actual starting position \(\left[ W_{0}\; C_{0}\; H_{0} \right] \) of the group we are considering, whereas the state dependence vector \(\left[ W_{0}^{\prime }\; C_{0}^{\prime }\; H_{0}^{\prime } \right] \) is determined by the counterfactual starting position of the destination group. The same procedure is used for the other four dependent variables.

Now, if we denote the life expectancy calculated according to this counterfactual experiment by LE , we could calculate the expected gain from moving from one starting position to another one as LE  − LE. We have called this difference ‘gain’ in the paper. Obviously, this number can be larger or smaller than the difference ΔLE defined above.

Example of a Life Table

Below we present two examples of the life tables which can be produced based on the parameter estimates from Section “Results”. In Table 20, we compare two males who are identical in all respects except for their educational attainment. In the left hand column, we present a life table for an individual with a university degree, and in the right column the corresponding table for someone without any education.

Table 20 Life tables for healthy cohabiting and working males with and without university degrees

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Karlsson, M., Mayhew, L. & Rickayzen, B. Individualised Life Tables. Population Ageing 1, 153–191 (2008). https://doi.org/10.1007/s12062-009-9008-2

Download citation

Keywords

  • Disability
  • Cohabitation
  • Mortality
  • Labour supply
  • Maximum simulated likelihood
  • Attrition